Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22

The sine shaped yarn further is presumed to be fixed at x 0  . The displacements are applied at the zero-crossing x L  . A comparable value of deformation due to a displacement in x -direction is the sum of the original length l L 0  and the applied displacement l l l l L 1 0 1      referred to the original length by

l     

l L l L L L 1 1  l

 

u

(7)

1

rel

l

0

that is positive for elongation and negative for shortening. The above stated and introduced relation of relative displacements u rel according to Eq. (7) is not a classic elongation or compression that leads to stresses as internal forces, but a geometric deformation due to displacements applied on an ideally stiff and at the same time ideally flexible yarn, indicated by the relations (6) that idealize the qualitative relation (1).

1,5 .

%10 %20

u u u

  

rel

rel

1,0 .

%1

rel

%1

u u u

  

rel

%20 %10

0,5 .

rel

rel

0,0 .

 2 sin L x A xy   

  

urel=0   

mit

 2 ,1

L A

 

0

-0,5 .

     

     

x

 

urel=+0,2 sin 1  

für

0

A xy 

u







rel

1

rel u L



-1,0 .

x

 

2 sin A xy 

für

0

u

urel=-0,2 1  







rel

1

rel u L



-1,5 .

0,0 0

1,6 π/2

3,1 π

4,7 3π/2

6,3 2π

Figure 2 : Obtained sines under the presumption of purely geometric deformation of an initial sine with A 0 graph as bold solid line; Elongated graphs as dashed lines; Shortened graphs as dash-dotted lines.

=1 and L =2π: Original

The applied degrees of deformation u rel

(7) have been selected in relevant ranges for structural mechanics of fiber





3

3

in steps

reinforced plastics [10, 30-33]. In detail two decades are considered. The large interval is rel u

  

  

1 10

1 10





of 4 1 10   and the small interval is rel u

5 1 10   . Counting the zero 39 and 21 substeps

4

4

in steps of

  

  

1 10

1 10

result for u rel .

The previously stated presumptions lead to two different effects in the model. Positive deformations lead to a smoothing or flattening. The amplitude decreases. In this case the maximum in elongation is reached when the yarn gets completely flattened. In contrast the amplitude increases for a negative deformation applied. In this case the maximum applicable value of the negative deformation is the limit of the domain of definition. Fig. 2 shows the obtained sines under the presumption of purely geometric deformation of an initial sine with an initial amplitude A 0 1  and domain of the argument   D 0, 2   .

Mathematical processing The arc length of a function is defined by [27, 28]

L

y dx 2

  1    

(8)

s

0

As carried out in the following, in the case of the presumed sine the solution of Eq. (8) can be achieved by converting it into an elliptic integral of the second kind

232